Diametral diameter two properties, Daugavet-, and Δ-points in Banach spaces
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2020-07-10
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Abstract
Banachi ruumidel vaadeldavate geomeetriliste omaduste seas on äärmuslik ja hästi uuritud Daugaveti omadus. Sellise omadusega Banachi ruumi veidruseks on, et ühikkera iga viilu diameeter on 2 ehk tal on diameeter-2 omadus. Näiteks c_0, l_∞, C[0,1] ja L_1 [0,1] on sellised, teiselt poolt aga refleksiivsete sh Hilberti ruumide või separaablite kaasruumide nagu l_1 ühikkeras leidub kui tahes väikese diameetriga viile. Klassikaliste diameeter-2 omaduste uuringud pakuvad välja omaduste teisendeid, mida väitekirjas põhjalikult käsitletakse. Töö üheks lähtekohaks on uurimus, kus vaadeldi Banachi ruume, mille iga normiga 1 element on Δ-punkt, st mille igal ühikkera viilu elemendil leidub selles viilus peaaegu diametraalne vastaspunkt. Veidi teistsuguse tuntud geomeetrilise kirjelduse kohaselt on Banachi ruumil Daugaveti omadus, kui vabalt fikseeritud normiga 1 element x on Daugaveti-punkt, st iga selle ruumi ühikkera element on selles ruumis ligikaudu lähendatav elemendi x peaaegu diametraalsete punktide kumerate kombinatsioonidega. Väitekirjas selgitatakse välja uute diametraalsete diameeter-2 omaduste omavaheline vahekord, käsitletakse nende stabiilsust summa- ja komponentruumide vahel ja alamruumi pärandumist. Lisaks esitletakse näiteid tuntud Banachi ruumidest, kus Daugaveti- ja Δ-punktid on samad. Põhjalik ülevaade nende punktide olemasolust absoluutse normiga summaruumides näitab muuhulgas, et mõisted Daugaveti-punkt ja Δ-punkt on erinevad. Märgime lõpetuseks, et töös esineva olulise hulga tähisena on sümboli Δ-valik vähemalt osaliselt tribuut Tartu Ülikooli uuele Delta keskusele.
The Daugavet property is an extreme and well researched geometric property of Banach spaces. The peculiarity of Banach spaces with such property is that every slice of the unit ball is 2, i.e. that space has the diameter 2 property. Examples of such spaces are c_0, l_∞, C[0,1] ja L_1 [0,1]. Reflexive spaces, e.g. Hilbert spaces, or separable dual spaces, e.g. l_1, however, have slices of arbitrarily small diameter. The research on the classical diameter 2 properties suggests different versions of these properties and these are thoroughly studied in this thesis. One of the starting points of the thesis is a study, that considered Banach spaces in which every element of norm 1 is a Δ-point, that is, in which every element in a slice of the unit ball has an almost diametral point in that slice. According to a slightly different geometric characterisation a Banach space has the Daugavet property if every element x of norm 1 is a Daugavet-point, that is, every element of the unit ball can be approximated in that space with convex combinations of almost diametral points of x. In the current thesis, the relations between new diametral diameter 2 properties are explained, the stability and inheritance results of these properties are investigated. In addition, examples are presented of well-known Banach spaces where Daugavet- and Δ-points are the same. A thorough study of the existence of Daugavet- and Δ-points in absolute sums affirms, among other things, that the concepts Daugavet-point and Δ-point are different. Let it be remarked, that the choice of the symbol Δ to denote an important notion in the thesis is at least a partial tribute to the new Delta Centre of University of Tartu.
The Daugavet property is an extreme and well researched geometric property of Banach spaces. The peculiarity of Banach spaces with such property is that every slice of the unit ball is 2, i.e. that space has the diameter 2 property. Examples of such spaces are c_0, l_∞, C[0,1] ja L_1 [0,1]. Reflexive spaces, e.g. Hilbert spaces, or separable dual spaces, e.g. l_1, however, have slices of arbitrarily small diameter. The research on the classical diameter 2 properties suggests different versions of these properties and these are thoroughly studied in this thesis. One of the starting points of the thesis is a study, that considered Banach spaces in which every element of norm 1 is a Δ-point, that is, in which every element in a slice of the unit ball has an almost diametral point in that slice. According to a slightly different geometric characterisation a Banach space has the Daugavet property if every element x of norm 1 is a Daugavet-point, that is, every element of the unit ball can be approximated in that space with convex combinations of almost diametral points of x. In the current thesis, the relations between new diametral diameter 2 properties are explained, the stability and inheritance results of these properties are investigated. In addition, examples are presented of well-known Banach spaces where Daugavet- and Δ-points are the same. A thorough study of the existence of Daugavet- and Δ-points in absolute sums affirms, among other things, that the concepts Daugavet-point and Δ-point are different. Let it be remarked, that the choice of the symbol Δ to denote an important notion in the thesis is at least a partial tribute to the new Delta Centre of University of Tartu.
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Banach spaces